Comparing Grothendieck-Witt Groups of a Complex Variety to its Real Topological K-Groups
On a complex variety X, two different approaches to K-theory are available: the algebraic K-theory of the variety, and the topological K-theory of the underlying topological space. In this context, the algebraic variant known as Hermitian K-theory corresponds to topological KO-theory. Our aim is to compare the two approaches. We start by constructing a comparison map from certain Hermitian K-groups of X to the KO-groups of X. It is clear what this map must be on groups in degree zero, but the definitions of relative and higher groups differ widely in the algebraic and the topological setting. This difficulty can be overcome by viewing relative and higher groups as subgroups of degree zero groups of certain auxiliary spaces. Once the definition of our comparison map is in place, we prove a number of fundamental properties, in particular compatibility with pushforwards along closed embeddings. We also show how we can use it to compare an exact sequence relating usual algebraic K-theory to Hermitian K-theory with a portion of the Bott sequence in topology. This finally allows us to deduce that the map is an isomorphism on smooth cellular varieties. We conclude with some details concerning projective spaces, for which independent computations of the algebraic and the topological groups exist.